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17,636 published problems•46 topics•Updated 18 Oct 2026

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Year 12 Extension 1
6 mk

Integration and Differentiation of a Trigonometric Function

Given f(x)=sin⁡x+cos⁡xf(x)=\sin x+\cos x, find its derivative, horizontal tangents on [0,π][0,\pi], area under the curve from 00 to π2\frac{\pi}{2}, and volume of revolution.

Further Calculus

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Year 10

Mathematics

Numbers, algebra, geometry, measurement, statistics and probability — covering NSW Stage 5.1 / 5.2 / 5.3.

2,631 problems

Functions And Graphs · Algebra And Linear Relationships · Indices And Surds

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Further Calculus1989Financial Mathematics1252Probability939Functions908Statistics876Differentiation709Trigonometry664Complex Numbers510Proof By Induction419Mechanics397Differential Equations290Integration Techniques290

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Year 12 Extension 2
13 mk

A Chord Product Inequality and Sharp Logarithmic Bounds via Integration

A geometric construction is used to prove a two-variable product inequality, and integration of a related rational function then provides sharp bounds for log⁡2\log 2.

Inequalities
Year 12 Extension 2
10 mk

A Circle from a Real-Part Condition

Investigate the locus defined by Re⁡(z−2z+1)=0\operatorname{Re}\left(\frac{z-2}{z+1}\right)=0, determine its Cartesian equation and sketch it, then examine a family of such loci and an intersection problem.

Complex Numbers
Year 12 Extension 2
12 mk

A Circular Locus from a Reciprocal Argument Condition

Determine and describe the locus of points zz satisfying argleft(dfracz−2iz+2right)=dfracpi4\\arg\\left(\\dfrac{z-2i}{z+2}\\right)=\\dfrac{\\pi}{4}, then investigate a family of reciprocal-related loci.

Complex Numbers
Year 12 Extension 2
10 mk

A Condition for a Parallelogram to be a Square

For a parallelogram OABCOABC with vertices 0,w,w+z,z0,w,w+z,z, the condition z−w=ki(z+w)z-w=ki(z+w) for a real constant kk is explored. It is shown that k=1k=1 gives a square, and k≠1k\neq1 gives a rectangle whose side length depends on kk.

Complex Numbers
Year 12 Extension 2
11 mk

A Cosh–Sinh Inequality via the Arithmetic–Geometric Mean

Prove the inequality cosh⁡xcosh⁡y≥cosh⁡2 ⁣(x+y2)\cosh x\cosh y \ge \cosh^2\!\left(\frac{x+y}{2}\right) by reducing it to the arithmetic–geometric mean inequality for positive quantities, then generalise to an nn-term product.

Inequalities
Year 12 Extension 2
11 mk

A Cosine Small-Angle Bound and a New Factorial Double Inequality

Prove the bound cos⁡x≤1−x27\cos x \le 1-\frac{x^2}{7} on [0,π/2][0,\pi/2], then apply it to create trigonometric products whose sizes are squeezed by a factorial inequality.

Inequalities
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